
theorem Th4:
  for L being non empty addLoopStr, p being Element of (the
  carrier of L)* holds <*Sum p*> = Sum<*p*>
proof
  let L be non empty addLoopStr, p be Element of (the carrier of L)*;
A1: now
    let i be Nat;
    assume i in dom<*p*>;
    then i in {1} by FINSEQ_1:2,38;
    then
A2: i = 1 by TARSKI:def 1;
    hence <*Sum p*>/.i = Sum p by FINSEQ_4:16
      .= Sum(<*p*>/.i) by A2,FINSEQ_4:16;
  end;
A3: dom <*Sum p*> = Seg 1 by FINSEQ_1:38
    .= dom <*p*> by FINSEQ_1:38;
  then len <*Sum p*> = len <*p*> by FINSEQ_3:29;
  hence thesis by A3,A1,MATRLIN:def 6;
end;
